Observing Quantum Monodromy An Energy-Momentum Map Built From Experimentally-Determined Level Energies Obtained from the ν7 Far-Infrared Band System of NCNCS Dennis W. Tokaryk, Stephen C. Ross Department of Physics and Centre for Laser, Atomic, and Molecular Sciences University of New Brunswick, Fredericton, NB Canada Brenda P. Winnewisser, Manfred Winnewisser, Frank C. De Lucia Department of Physics, The Ohio State University, Columbus, OH USA Brant E. Billinghurst Canadian Light Source, Inc., Saskatoon, SK Canada MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 1

Monodromy – you’ve seen it before! A simple example from complex analysis: let z = x + iy = r eiθ. Then the function, ln(z) = ln(r) + iθ, has a singularity at z = 0 + i0 = 0 eiθ! (because ln(0) is undefined.) Re(z) Im(z) z y r θ x MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 2

Monodromy – you’ve seen it before! Make a circuit around the singularity, evaluating ln(z) en route: At the start of the circuit θ = 0, so ln(z) = ln(r0) + i0 = ln(r0). Re(z) Im(z) Start here ln(z) = ln(r0) z θ r0 MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 3

Monodromy – you’ve seen it before! At the end of the circuit, θ = 2πi, so ln(z) = ln(r0) + 2πi. The value of ln(z) is not single-valued: - the value depends on: how we got to z (i.e. upon its history). Im(z) θ Re(z) r0 z End here ln(z) = ln(r0) + 2πi MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 4

quasi-linear molecules Chemistry Molecules (synthesis) Physics/ Technology Spectroscopic data Quantum monodromy as seen in quasi-linear molecules Theoretical spectroscopy GSRB Hamiltonian Topology of the phase-space surfaces of constant E, J Mathematics MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 5

Timeline of LAM theoretical and experimental spectroscopy Year 1970 The Vibration-Rotation Problem in Triatomic Molecules Allowing for a Large-Amplitude Bending Vibration, HBJ = Hougen, Bunker and Johns, J. Mol. Spectrosc. 34, 136 (1970) The effective rotation-bending Hamiltonian of a triatomic molecule, and its application to extreme centrifugal distortion in the water molecule, Hoy and Bunker, J. Mol. Spectrosc. 52, 439 (1974) Semi-rigid bender (SRB) Hamiltonian by Bunker and Landsberg, J. Mol. Spectrosc. 67, 374 (1977) 1980 A reinterpretation of the CH2 photoelectron spectrum, Sears and Bunker, J. Chem. Phys. 79, 5265 (1983) Analysis of the laser photoelectron spectrum of CH2, Bunker and Sears, J. Chem. Physics 83, 4866 (1985): We NOW know that an energy-momentum map of their calculated rotation-bending energies in their Table V shows quantum monodromy! 1990 Morse Oscillator Rigid Bender Internal Dynamics [MORBID], Jensen, J. Mol. Spectrosc. 128, 478 (1988). OCCCS, NCNCS, NCNCO, and NCNNN as Semirigid Benders, [Hamiltonian now called: Generalized Semi-Rigid Bender (GSRB) ], Ross, J. Mol. Spectrosc. 132, 48 (1988) Experimental confirmation of quantum monodromy: The millimeter wave spectrum of cyanogen isothiocyanate NCNCS, B. and M. Winnewisser, Medvedev, Behnke, DeLucia, Ross and Koput, PRL, 243002 (2005) 2000 The hidden kernel of molecular quasi-linearity: Quantum monodromy, M. and B. Winnewisser, Medvedev, DeLucia, Ross, Bates, J. Mol. Structure 798, 1-26 (2006) Analysis of the FASSST rotational spectrum of NCNCS in view of quantum monodromy, B. and M. Winnewisser, Medvedev, DeLucia, Ross and Koput, PCCP, 12, 8158 ( 2010) 2010 Pursuit of quantum monodromy in the far-infrared and mid-infrared spectra of NCNCS using synchrotron radiation, M. and B. Winnewisser, DeLucia, Tokaryk, Ross and Billinghurst, PCCP, 16, 17373 (2014) MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 6

Timeline of Mathematics leading to Monodromy in Molecules Year Timeline of Mathematics leading to Monodromy in Molecules 1970 Geometrical obstructions to global action-angle variables, Nekhoroshev, Trans. Moskow Math. Soc. 26, 180 (1972) On global action-angle coordinates, Duistermaat, Comm. Pure and Appl. Math. 33, 687 (1980) 1980 The quantum mechanical spherical pendulum, Cushman and Duistermaat, Bull. Am. Math. Soc. 19, 475 (1988) Monodromy in the champagne bottle, Bates, J. Appl. Math. and Physics 42, 837 (1991) Classical energy-momentum map 1990 Quantum states in a champagne bottle, Child, J. Phys. A: Math. Gen. 31, 657 (1998) Energy-momentum map of discrete quantum levels Quantum monodromy in the spectrum of HOH and other systems: new insight into the level structure of quasi-linear molecules, Child, Weston and Tennyson, Mol. Phys. 96, 371 (1999) 2000 Monodromy in the water molecule, Zobov, Shirin, Polyansky, Tennyson, Coheur, Bernath, Chemical physics letters, 414, 193 (2005): Comparison of high temperature HOH data with predictions Hamiltonian monodromy as lattice defect, Zhilinskii, Topology in condensed matter, Springer series: Vol. 150, 186 (2006) 2010 Quantum monodromy and molecular spectroscopy, Child, Contemporary Physics 55, 212 (2014) MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 7

Classical motion in the champagne-bottle potential The classical motion was studied in 1991 by Larry Bates. (Bates, J. Ap. Math. Physics 42 (1991) 837) He made an Energy-Momentum map and identified the ‘critical points’ at which the radial momentum is zero. These form the parabolic curve + the critical point (k = 0, e = the energy of the top of the potential energy hump) shown here: k: angular momentum e: total energy Critical points Energy-Momentum map MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 8

Classical motion in the champagne-bottle potential The classical action variable for radial motion is, Ir ~ the frequency of a particle’s radial oscillation (for the given energy e and angular momentum k). • Evaluate Ir at a starting point on the loop shown on the energy-momentum map. • Move a small distance around the loop, re-evaluate Ir. • Continue all the way around the loop, - calculate Ir at each point. • Result: Ir has a different value when you return to the starting point! k: angular momentum e: total energy Ir MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 9

Classical motion in the champagne-bottle potential Our classical action experiences a branch cut as shown! So, the vibrational periodicity Ir represents will not smoothly change when crossing over this line. Branch cut k: angular momentum e: total energy MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 10

Monodromy! Monodromy! Larry Bates (University of Calgary) GSRB: Quantum Hamiltonian Monodromy in the champagne bottle J. Appl. Math. and Physics 42, 837 (1991) 2 1 µρρJρ2 2 1 µzzJz2 H = + V(ρ) + Many more terms + = Tvib + Tz-rot + Classical Hamiltonian Many more terms + pR 2 j 2 H = + + V(R) 2m 2mR2 GSRB contains the essential core which leads to… = Tvib + Tz-rot + Monodromy! Monodromy! • GSRB already accounted for Monodromy before we even knew it existed! MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 11

Quantum motion in the champagne-bottle potential Mark Child quantized the classical problem, and made an energy-momentum map of discrete quantum levels. (Child, J. Phys. A: Math. Gen. 31 (1998) 657; Contemporary Physics 55 (2014) 212) quantum level of energy E and angular momentum k. line of constant radial vibration v = 1 v = 0 MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 12

a-type sub-bands b-type sub-bands b-type sub-bands Spectra of the ν7 low-frequency high-amplitude bending mode and of the of the ν3 hybrid band stretching mode, taken at the CLS. ~68 a-type sub-bands of the ν7 mode (For experimental details, see Winnewisser et al, PCCP Perspectives 16, 17373 (2014 ) ) a-type sub-bands b-type sub-bands b-type sub-bands MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 13

The energy-momentum map How we connected Ka stacks and different vibrational levels with data from the CLS Up to Ka = 10 Ka = 10 Ka = 12 Ka = 12 MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 14

The NCNCS energy-momentum map showing the quantum lattice for the ν7 in-plane large amplitude bending mode Small Dots: show J = Ka levels calculated from the analysis of the pure rotational spectrum only Centers of the Red Circles: show levels measured directly from the assignment of the FIR spectrum MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 15

Analysis of the laser photo-electron spectrum of CH2, Bunker and Sears, J. Chem. Phys. 83 4866(1985) We NOW know that an energy- momentum map of the calculated rotation-bending energies in their Table V shows quantum monodromy MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 16

Monodromy plot for water Child, Contemporary Physics, 55 212 (2014) with data from Zobov et. al., 414 193 (2005). MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 17

• We determined precise relative energies for the lowest seven The take-home part… • We determined precise relative energies for the lowest seven vibrational levels of the ν7 mode of NCNCS: for Ka up to (a maximum of) 12. • Our energy-momentum map contains all of the structure due to quantum monodromy expected for a quasi-linear molecule. • Stephen Ross’ GSRB Hamiltonian proved excellent at predicting the positions of these energy levels, even though only pure rotational spectra were included in the fitting. • If you are working on a new quasi-linear molecule, an energy-momentum map will be a helpful aid to determining the height of the barrier to linearity, since the general structure of the map will be the same for every such molecule. MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 18

• Natural Sciences and Engineering Research Council of Canada (NSERC) Thanks • Natural Sciences and Engineering Research Council of Canada (NSERC) – Discovery grants for Drs. Ross and Tokaryk • Damien Forthomme and Colin Sonnichsen for help with the experiments • Staff at the CLS for technical support and accommodation during the experiments. MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 19

Classical motion in the champagne-bottle potential Mark Child explains why not. (Child, J. Phys. A: Math. Gen. 31 (1998) 657; Contemporary Physics 55 (2014) 212) For small values of k > 0, between successive radial maxima, Δθ > 0, but very small. .. but now, for the same initial conditions, Δθ ~ +π, and direction of rotation reverses! Child shows that the derivative of the action wrt k is negative if k>0, and positive if k<0… discontinuous at k = 0! won’t depend on k (same radial vibrational frequency for all small k, positive or negative.) MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 20

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MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 22

Quantum motion in the champagne-bottle potential What did our game mean?? That the quantum numbers v and k, which we took as valid below the monodromy point, were not universal. New quantum numbers of vibration and angular momentum are necessarily required above the monodromy point. MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 23

Quantum motion in the champagne-bottle potential Rules of the game: Define a closed path γ through a set of quantum levels circling the monodromy point. From each point, increase k by one unit, then increase v by one unit. Make a closed trapezium. The base of the previous trapezium should not make a drastic change of orientation from the previous one. Check the return point – once again, your horse turns into a cow! MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, 2016 Slide 24